Uniform boundedness principle

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

Theorem

Theorem (Uniform Boundedness Principle). Let X be a Banach space and Y be a normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. If for all x in X one has

\sup\nolimits_{T \in F} \|T(x)\|_Y < \infty,

then

\sup \nolimits _{{T\in F,\|x\|=1}}\|T(x)\|_{Y}=\sup \nolimits _{{T\in F}}\|T\|_{{B(X,Y)}}<\infty .

Proof

The completeness of X enables the following short proof, using the Baire category theorem.

Proof. Suppose that for every x in the Banach space X, one has:

\sup\nolimits_{T \in F} \|T (x)\|_Y < \infty.

For every integer $n\in \mathbb {N}$ , let

X_n = \left \{x \in X \ : \ \sup\nolimits_{T \in F} \|T (x)\|_Y \le n \right \}.

The set $X_{n}$ is a closed set and by the assumption,

\bigcup\nolimits_{n \in \mathbf{N}} X_n = X \neq \varnothing.

By the Baire category theorem for the non-empty complete metric space X, there exists m such that $X_m$ has non-empty interior, i.e., there exist $x_0 \in X_m$ and ε > 0 such that

\overline{B_\varepsilon (x_0)} := \{x \in X \,:\, \|x - x_0\| \le \varepsilon \} \subseteq X_m.

Let u ∈ X with ǁuǁ ≤ 1 and T ∈ F. One has that:

\begin{align} \|T(u) \|_Y &= \varepsilon^{-1} \left \|T \left( x_0 + \varepsilon u \right) - T(x_0) \right \|_Y & [\text{by linearity of } T ] \\ &\leq \varepsilon^{-1} \left ( \left\| T (x_0 + \varepsilon u) \right\|_Y + \left\| T (x_0) \right\|_Y \right ) \\ &\leq \varepsilon^{-1} (m + m). & [ \text{since} \ x_0 + \varepsilon u, \ x_0 \in X_m ] \\ \end{align}

Taking the supremum over u in the unit ball of X, it follows that

\sup\nolimits_{T \in F} \|T\|_{B(X,Y)} \leq 2 \varepsilon^{-1} m < \infty.

There are also simple proofs not using the Baire theorem (Sokal 2011).

Corollaries

Corollary. If a sequence of bounded operators (T_n) converges pointwise, that is, the limit of {T_n(x)} exists for all x in X, then these pointwise limits define a bounded operator T.

Note it is not claimed above that T_n converges to T in operator norm, i.e. uniformly on bounded sets. (However, since {T_n} is bounded in operator norm, and the limit operator T is continuous, a standard "3-ε" estimate shows that T_n converges to T uniformly on compact sets.)

Corollary. Any weakly bounded subset S in a normed space Y is bounded.

Indeed, the elements of S define a pointwise bounded family of continuous linear forms on the Banach space X = Y*, continuous dual of Y. By the uniform boundedness principle, the norms of elements of S, as functionals on X, that is, norms in the second dual Y**, are bounded. But for every s in S, the norm in the second dual coincides with the norm in Y, by a consequence of the Hahn–Banach theorem.

Let L(X, Y) denote the continuous operators from X to Y, with the operator norm. If the collection F is unbounded in L(X, Y), then by the uniform boundedness principle, we have:

R = \left \{ x \in X \ : \ \sup\nolimits_{T \in F} \|Tx\|_Y = \infty \right \} \neq \varnothing

In fact, R is dense in X. The complement of R in X is the countable union of closed sets ∪X_n. By the argument used in proving the theorem, each X_n is nowhere dense, i.e. the subset ∪X_n is of first category. Therefore R is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows:

Theorem. Let X be a Banach space, {Y_n} a sequence of normed vector spaces, and F_n a unbounded family in L(X, Y_n). Then the set

R = \left \{ x \in X \ : \ \forall n \in \mathbf{N} : \sup\nolimits_{T \in F_n} \|Tx\|_{Y_n} = \infty \right \}

is of second category, and thus dense in X.

Proof. The complement of R is the countable union

\bigcup\nolimits_{n,m} \left \{ x \in X \ : \ \sup\nolimits_{T \in F_n} \|Tx\|_{Y_n} \le m \right \}

of sets of first category. Therefore its residual set R is dense.

Example: pointwise convergence of Fourier series

Let $\mathbb {T}$ be the circle, and let $C(\mathbb{T})$ be the Banach space of continuous functions on $\mathbb {T}$ , with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in $C(\mathbb{T})$ for which the Fourier series does not converge pointwise.

For $f \in C(\mathbb{T})$ , its Fourier series is defined by

\sum_{k \in \mathbf{Z}} \hat{f}(k) e^{ikx} = \sum_{k \in \mathbf{Z}}\frac{1}{2\pi} \left (\int_0 ^{2 \pi} f(t) e^{-ikt} dt \right) e^{ikx},

and the N-th symmetric partial sum is

S_N(f)(x) = \sum_{k=-N}^N \hat{f}(k) e^{ikx} = \frac{1}{2 \pi} \int_0 ^{2 \pi} f(t) D_N(x - t) \, dt,

where D_N is the N-th Dirichlet kernel. Fix $x \in \mathbb{T}$ and consider the convergence of {S_N(f)(x)}. The functional φ_N,x : $C(\mathbb{T})\rightarrow \mathbb{C}$ defined by

\varphi_{N, x} (f) = S_N(f)(x), \qquad f \in C(\mathbb{T}),

is bounded. The norm of φ_N,x, in the dual of $C(\mathbb{T})$ , is the norm of the signed measure (2π)⁻¹D_N(x−t) dt, namely

\left\|\varphi _{{N,x}}\right\|={\frac {1}{2\pi }}\int _{0}^{{2\pi }}\left|D_{N}(x-t)\right|\,dt={\frac {1}{2\pi }}\int _{0}^{{2\pi }}\left|D_{N}(s)\right|\,ds=\left\|D_{N}\right\|_{{L^{1}({\mathbb {T}})}}.

One can verify that

\frac{1}{2 \pi} \int_0 ^{2 \pi} | D_N(t) | \, dt \ge \int_0^\pi \frac{\left |\sin\left ( (N+\tfrac{1}{2})t \right )\right|} t \, dt \to \infty.

So the collection {φ_N,x} is unbounded in $C(\mathbb{T})^\ast$ , the dual of $C(\mathbb{T})$ . Therefore by the uniform boundedness principle, for any $x \in \mathbb{T}$ , the set of continuous functions whose Fourier series diverges at x is dense in $C(\mathbb{T})$ .

More can be concluded by applying the principle of condensation of singularities. Let {x_m} be a dense sequence in $\mathbb {T}$ . Define φ_{N,x_m} in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each x_m is dense in $C(\mathbb{T})$ (however, the Fourier series of a continuous function f converges to f(x) for almost every $x \in \mathbb{T}$ , by Carleson's theorem).

Generalizations

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):

Theorem. Given a barrelled space X and a locally convex space Y, then any family of pointwise bounded continuous linear mappings from X to Y is equicontinuous (even uniformly equicontinuous).

Alternatively, the statement also holds whenever X is a Baire space and Y is a locally convex space (Shtern 2001).

Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically,

Theorem. Let X be a Fréchet space, Y a normed space, and H a set of continuous linear mappings of X into Y. If for every x in X

\sup\nolimits_{u\in H}\|u(x)\|<\infty,

then the family H is equicontinuous.

References

Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF), Fundamenta Mathematicae, 9: 50–61 . (French)
Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Springer, ISBN 978-3-540-42338-6
Dieudonné, Jean (1970), Treatise on analysis, Volume 2, Academic Press .
Rudin, Walter (1966), Real and complex analysis, McGraw-Hill .
Shtern, A.I. (2001), "b/b015200", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .
Sokal, Alan (2011), "A really simple elementary proof of the uniform boundedness theorem", Amer. Math. Monthly, 118: 450–452, arXiv:1005.1585, doi:10.4169/amer.math.monthly.118.05.450 .

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